2007/01/03 by Helge Holden, Kenneth H. Karlsen, Nils Henrik Risebro · 28 citations
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Advanced Mathematical Physics Problems #Advanced Differential Equations and Dynamical Systems #Mathematics #Dissipative system #Convergence (economics) #Finite difference #Applied mathematics #Numerical analysis #A priori and a posteriori #Mathematical analysis #Physics
paper · pdf · doi:10.1090/s0025-5718-07-01919-9
published in Mathematics of Computation 76(258), 699-744 (American Mathematical Society)
openalex publication_date 2007/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/23
We propose and analyze several finite difference schemes for the HunterâSaxton equation ut+u ux= \frac 12 ∫ 0x (ux)2 dx, x>0, t>0. This equation has been suggested as a simple model for nematic liquid crystals. We prove that the numerical approximations converge to the unique dissipative solution of (HS), as identified by Zhang and Zheng. A main aspect of the analysis, in addition to the derivation of several a priori estimates that yield some basic convergence results, is to prove strong convergence of the discrete spatial derivative of the numerical approximations of u, which is achieved by analyzing various renormalizations (in the sense of DiPerna and Lions) of the numerical schemes. Finally, we demonstrate through several numerical examples the proposed schemes as well as some other schemes for which we have no rigorous convergence results.